which of the following correlation demonstrate a casual relationship?

1 the distance student live from school and the time it takes each to walk there.

2 the amount of money received for allowance and the number of DVD's purchased.

3 the number of hours spend studying and the grades received on exams.

4 high school students' ages and the number of hours they sleep.

Do you mean CASUAL or CAUSAL?

i did mean casual

Then, with a casual relationship, all but 1 are correct.

To determine if a correlation demonstrates a causal relationship, you need to establish a few criteria. While correlations can suggest a relationship between two variables, it doesn't necessarily mean that one variable causes the other. Below are the criteria that need to be met for a causal relationship:

1. Temporal Relationship: The cause must precede the effect in time. In other words, changes in the cause must occur before changes in the effect.

2. Correlation: There must be a correlation between the cause and effect. This means that as the cause variable changes, the effect variable also changes consistently.

3. Elimination of Alternative Explanations: Any other plausible explanations for the observed relationship must be eliminated. This is typically done through careful research design, controlling relevant variables, and ruling out confounding factors.

4. Coherence with Existing Knowledge: The causal relationship should be consistent with prior knowledge and theories in the field.

Analyzing the given options:

1. The distance students live from school and the time it takes them to walk there:
- Temporal Relationship: It is likely that the distance students live from school precedes the time it takes to walk there.
- Correlation: There is a possibility of a correlation if students living closer to the school tend to have shorter walking times.
- Elimination of Alternative Explanations: Other factors like physical fitness, walking speed, or availability of transportation could affect the time to walk to school.
- Coherence with Existing Knowledge: This relationship is consistent with our understanding of commuting distances and required walking times.

2. The amount of money received for allowance and the number of DVDs purchased:
- Temporal Relationship: It is possible that the amount of money received for allowance precedes the purchase of DVDs.
- Correlation: There might be a correlation if individuals with higher allowances tend to buy more DVDs.
- Elimination of Alternative Explanations: Other factors, such as individual preferences, availability of other entertainment options, or media consumption habits, can also influence DVD purchases.
- Coherence with Existing Knowledge: This relationship is consistent with the idea that having more money allows for more spending on leisure activities.

3. The number of hours spent studying and the grades received on exams:
- Temporal Relationship: It is reasonable to assume that studying precedes taking the exams and receiving grades.
- Correlation: There is a possibility of a correlation as more studying could result in higher grades.
- Elimination of Alternative Explanations: Other factors like intelligence, prior knowledge, study techniques, or external distractions may impact exam grades.
- Coherence with Existing Knowledge: This relationship aligns with the idea that more studying generally leads to better performance on exams.

4. High school students' ages and the number of hours they sleep:
- Temporal Relationship: Age (in this case, chronological age) naturally precedes the number of hours slept.
- Correlation: There might be a correlation if younger students generally need more sleep.
- Elimination of Alternative Explanations: Other factors, such as lifestyle choices, health conditions, or personal habits, can also influence the number of hours slept.
- Coherence with Existing Knowledge: Younger individuals generally require more sleep, and sleep needs tend to decrease as people grow older.

Considering the above analysis, option 4 (high school students' ages and the number of hours they sleep) seems to meet the criteria for a correlation that demonstrates a causal relationship. However, it is important to note that establishing a true causal relationship often requires rigorous scientific research beyond basic correlations.